[César Lozada]:
Problem 6: Given triangle ABC with its circumcircle (O) and its orthocenter H. Let HaHbHc be
the orthic triangle of triangle ABC:P is an arbitrary point on Euler line. AP;BP;CP intersect (O)
again at A1;B1;C1: Let A2;B2;C2 be the reflections of A1;B1;C1 wrt Ha;Hb;Hc, respectively. Prove
that H;A2;B2;C2 are concyclic.
Source: Thành Viên - 111 Nice Geometry Problems from Mathscope (January 01,2013)
------------------------------------------------------------------------------
If OP = t*OH then the circle has radius OH*R^2*|(t-1)/(OH^2*t-R^2)| and center:
O* = (a^10-(b^2+c^2)*a^8-(2*b^4-7*b^2*c^2+2*c^4)*a^6+(2*b^2-c^2)*(b^2-2*c^2)*(b^2+c^2)*a^4+(b^4-4*b^2*c^2+c^4)*(b^2-c^2)^2*a^2-(b^4-c^4)*(b^2-c^2)^3)*t-a^2*b^2*c^2*(3*a^4-(b^2+c^2)*a^2-2*(b^2-c^2)^2) : : (barys)
O* lies on the Euler line and also:
· O* = reflection in O
of the inverse in the npc circle
of the inverse of P in the polar circle.
· OO*/OH= (2*(5*R^2-SW)*t-2*R^2)/(OH^2*t-R^2)
· O* has Shinagawa coefficients: (R^2*(1-t), (21*R^2-4*SW)*t-5*R^2)
ETC pairs (P,O*(P)): (3,382), (4,4), (22,18534), (23,11799), (24,3), (25,381), (26,7517), (378,3830), (403,18403), (468,7574), (1113,10750), (1114,10751), (1593,5076), (2070,2070), (3147,14790), (3515,1657), (3517,1656), (3518,5), (3520,3627), (3542,18404), (6353,18531), (7501,6923), (7505,18569), (7512,7553), (10295,18325), (10594,3843), (13596,15687), (14865,3853), (16868,18377), (20832,13743), (21213,12083), (21844,3146), (26863,3861)
Some others:
O*( X(2) ) = EULER LINE INTERCEPT OF X(66)X(265)
= a^10-(b^2+c^2)*a^8-2*(b^4+b^2*c^2+c^4)*a^6+2*(b^6+c^6)*a^4+(b^4-c^4)^2*a^2-(b^4-c^4)*(b^2-c^2)^3 : : (barys)
= R^2*S^2+(3*R^2-2*SW)*SB*SC : : (barys)
= 3*X(14561)-2*X(19127), 3*X(14643)-2*X(16165)
= As a point on the Euler line, this center has Shinagawa coefficients (-E, 5*E+8*F)
= lies on these lines: {2, 3}, {49, 9833}, {52, 18381}, {66, 265}, {68, 6243}, {115, 571}, {143, 18912}, {317, 339}, {399, 12319}, {497, 9642}, {511, 18474}, {568, 1899}, {570, 5475}, {578, 11750}, {1154, 11442}, {1352, 9019}, {1503, 18445}, {3060, 25739}, {3313, 3818}, {3521, 14542}, {3567, 18952}, {4857, 9644}, {5157, 19130}, {5448, 26883}, {5512, 15563}, {5654, 10540}, {5946, 18911}, {6033, 13556}, {6102, 11457}, {6776, 15087}, {7747, 19220}, {9927, 11572}, {10316, 27371}, {10539, 13419}, {10620, 13203}, {11392, 18447}, {11393, 18455}, {11433, 13321}, {11550, 13754}, {12295, 19506}, {13202, 19479}, {13352, 18400}, {13598, 18383}, {14516, 16266}, {14561, 19127}, {14643, 16165}, {14983, 19160}, {16655, 22660}, {18388, 29012}, {18390, 19161}, {18439, 22661}, {21850, 26926}, {22120, 27376}
= midpoint of X(4) and X(7391)
= reflection of X(i) in X(j) for these (i, j): (3, 427), (20, 18570), (22, 5), (7555, 13413), (12083, 15760), (14983, 19160)
= anticomplement of X(7502)
= orthocentroidal circle-inverse-of X(11818)
= {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (381, 12083, 15760), (382, 7517, 7553), (12225, 15559, 7526)
= [ 25.4691931202392600, 24.5324848228017200, -25.0983756816810900 ]
O*( X(5) ) = EULER LINE INTERCEPT OF X(49)X(18400)
= a^10-(b^2+c^2)*a^8-(2*b^4-b^2*c^2+2*c^4)*a^6+(b^2+c^2)*(2*b^4-3*b^2*c^2+2*c^4)*a^4+(b^4+c^4)*(b^2-c^2)^2*a^2-(b^4-c^4)*(b^2-c^2)^3 : : (barys)
= R^2*S^2+(11*R^2-4*SW)*SB*SC : : (barys)
= As a point on the Euler line, this center has Shinagawa coefficients (-E, 5*E+16*F)
= lies on these lines: {2, 3}, {49, 18400}, {50, 1879}, {52, 265}, {67, 17505}, {70, 21400}, {113, 13419}, {343, 12307}, {399, 22660}, {567, 3574}, {1141, 12092}, {1351, 18382}, {1352, 12061}, {1479, 9642}, {1568, 18350}, {1994, 20424}, {3060, 18394}, {3521, 10575}, {3581, 5449}, {3583, 9630}, {5012, 13470}, {5446, 13851}, {5448, 10540}, {5562, 6288}, {6102, 25739}, {6146, 15087}, {6241, 15134}, {6242, 12219}, {6243, 9927}, {6247, 10620}, {7728, 11381}, {10263, 18379}, {10733, 15132}, {11550, 18439}, {11572, 13754}, {11591, 22804}, {11750, 18388}, {12022, 14627}, {12062, 18387}, {12164, 22661}, {13556, 22823}, {15107, 23330}, {15432, 18428}, {18436, 18474}
= reflection of X(i) in X(j) for these (i, j): (3, 1594), (2937, 10024), (7488, 5), (10024, 23047)
= {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (381, 382, 7517), (3627, 18567, 4), (10750, 10751, 2070)
= [ 1.2448751260128830, 0.3720711911001485, 2.8085189837552540 ]
O*( X(20) ) = EULER LINE INTERCEPT OF X(52)X(22802)
= a^10-(b^2+c^2)*a^8-2*(b^4-5*b^2*c^2+c^4)*a^6+2*(b^4-3*b^2*c^2+c^4)*(b^2+c^2)*a^4+((b^2-c^2)^2-4*b^2*c^2)*(b^2-c^2)^2*a^2-(b^4-c^4)*(b^2-c^2)^3 : : (barys)
= R^2*S^2-(13*R^2-2*SW)*SB*SC : : (barys)
= As a point on the Euler line, this center has Shinagawa coefficients (E, -5*E+8*F)
= lies on these lines: {2, 3}, {52, 22802}, {68, 18439}, {113, 13346}, {184, 12897}, {265, 14216}, {388, 9642}, {399, 6193}, {1514, 22660}, {1533, 21659}, {2883, 18445}, {3521, 14457}, {5270, 9644}, {6000, 25738}, {6225, 18917}, {6243, 7728}, {9927, 11381}, {10540, 12118}, {10575, 18390}, {10620, 12250}, {11456, 12370}, {11663, 31670}, {12121, 20771}, {13445, 26917}, {13474, 18474}, {13491, 18912}, {13556, 22337}, {15041, 18933}, {15063, 15083}, {15072, 18952}, {17702, 26883}, {22661, 31383}
= midpoint of X(382) and X(7517)
= reflection of X(i) in X(j) for these (i,j): (3, 235), (11413, 5), (12121, 20771), (18404, 4), (31180, 3845)
= {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4, 3146, 18569), (3627, 7553, 382), (5073, 18403, 14790)
= [ -10.8678163907308700, -11.7086667395809700, 16.7625797896469900 ]
O*( X(30) ) = EULER LINE INTERCEPT OF X(113)X(22115)
= a^10-(b^2+c^2)*a^8-(2*b^4-7*b^2*c^2+2*c^4)*a^6+(2*b^2-c^2)*(b^2-2*c^2)*(b^2+c^2)*a^4+(b^4-4*b^2*c^2+c^4)*(b^2-c^2)^2*a^2-(b^4-c^4)*(b^2-c^2)^3 : : (barys)
= R^2*S^2-(21*R^2-4*SW)*SB*SC: : (barys)
= X(399)-4*X(1514), X(3581)+2*X(13202), X(13445)-3*X(14644)
= As a point on the Euler line, this center has Shinagawa coefficients (E, -5*E+16*F)
= lies on these lines: {2, 3}, {113, 22115}, {195, 5893}, {265, 6000}, {389, 3521}, {399, 1514}, {539, 15063}, {1154, 1539}, {1478, 9642}, {1495, 19479}, {3581, 13202}, {3585, 9627}, {4846, 16227}, {5878, 25738}, {6760, 18809}, {7728, 13417}, {7747, 18373}, {8718, 13470}, {9927, 18439}, {10113, 17854}, {10540, 17702}, {10620, 15311}, {10733, 14157}, {11455, 18392}, {11550, 18430}, {12295, 15089}, {13434, 15807}, {13445, 14644}, {13851, 14915}, {20127, 21663}, {20957, 22337}, {22816, 22951}
= midpoint of X(i) and X(j) for these {i, j}: {382, 2070}, {3146, 13619}, {10733, 14157}, {18325, 18403}
= reflection of X(i) in X(j) for these (i, j): (3, 403), (20, 15646), (186, 11563), (389, 13446), (858, 23323), (2070, 11799), (2071, 5), (2072, 10151), (6760, 18809), (7574, 18403), (11563, 11558), (13619, 7575), (18323, 13473), (18403, 4), (18859, 2072), (20127, 21663), (22115, 113), (25739, 10113)
= 2nd Droz-Farny circle-inverse-of X(5)
= polar circle-inverse-of X(18560)
= Stammler circle-inverse-of X(7517)
= {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4, 3146, 18377), (4, 18325, 7574), (15761, 18560, 3)
= [ -8.9552625482838670, -9.8011582624616960, 14.5592798397427200 ]
César Lozada
Δεν υπάρχουν σχόλια:
Δημοσίευση σχολίου